Objective

• To demonstrate, in the time domain, how a Linear Time-Invariant (LTI) system responds to a sinusoidal input.
• We will start from the convolution integral and use Euler’s identity to show that the steady-state output remains sinusoidal.

1. System Definition

A continuous-time LTI system is defined by:

\[ y(t) = x(t) * h(t) = \int_{-\infty}^{\infty} h(\tau)\,x(t - \tau)\,d\tau \]

where -\(x(t)\)is the input, -\(h(t)\)is the impulse response, -\(y(t)\)is the output.

2. Sinusoidal Input

Let the input be a sinusoid:

\[ x(t) = A \cos(\omega_0 t + \phi) \]

Using Euler’s identity:

\[ x(t) = \frac{A}{2} \left[e^{j(\omega_0 t + \phi)} + e^{-j(\omega_0 t + \phi)}\right] \]

3. Linearity of the Convolution Integral

Because the system is linear, we can treat each exponential term separately:

\[ y(t) = \frac{A}{2}\left[ y_1(t) + y_2(t) \right] \] where \[ y_1(t) = e^{j\phi} \int_{-\infty}^{\infty} h(\tau)\, e^{j\omega_0(t-\tau)}\, d\tau \] \[ y_2(t) = e^{-j\phi} \int_{-\infty}^{\infty} h(\tau)\, e^{-j\omega_0(t-\tau)}\, d\tau \]

4. Simplifying Each Integral

We can factor out the term\(e^{j\omega_0 t}\):

\[ y_1(t) = e^{j\phi} e^{j\omega_0 t} \int_{-\infty}^{\infty} h(\tau)\, e^{-j\omega_0 \tau}\, d\tau \]

Let \[ H_1 = \int_{-\infty}^{\infty} h(\tau)\, e^{-j\omega_0 \tau}\, d\tau \]

Then, \[ y_1(t) = e^{j\phi} H_1 e^{j\omega_0 t} \]

Similarly, \[ y_2(t) = e^{-j\phi} H_1^* e^{-j\omega_0 t} \]

5. Combine the Two Parts

The total output is:

\[ y(t) = \frac{A}{2}\left[e^{j\phi} H_1 e^{j\omega_0 t} + e^{-j\phi} H_1^* e^{-j\omega_0 t}\right] \]

If we express\(H_1\)in polar form: \[ H_1 = |H_1| e^{j\theta} \]

Then, \[ y(t) = A |H_1| \cos(\omega_0 t + \phi + \theta) \]

6. Interpretation in Time Domain

• The output is a cosine at the same frequency\(\omega_0\)as the input.
• Its amplitude is scaled by\(|H_1|\), which depends on\(h(t)\).
• Its phase is shifted by\(\theta\), the argument of\(H_1\).

\[ \boxed{y(t) = A |H_1| \cos(\omega_0 t + \phi + \theta)} \]

1. System Definition

A discrete-time LTI system is defined by the convolution sum:

\[ y[n] = \sum_{k=-\infty}^{\infty} h[k]\,x[n - k] \]

where - \(x[n]\) is the input sequence, - \(h[k]\) is the impulse response, - \(y[n]\) is the output sequence.

2. Sinusoidal Input

Let the input be a discrete sinusoid:

\[ x[n] = A \cos(\omega_0 n + \phi) \]

Using Euler’s identity:

\[ x[n] = \frac{A}{2}\left[e^{j(\omega_0 n + \phi)} + e^{-j(\omega_0 n + \phi)}\right] \]

3. Linearity of the Convolution Sum

Because the system is linear, each exponential term can be treated independently:

\[ y[n] = \frac{A}{2}\left[y_1[n] + y_2[n]\right] \]

where \[ y_1[n] = e^{j\phi} \sum_{k=-\infty}^{\infty} h[k]\, e^{j\omega_0 (n - k)} \] \[ y_2[n] = e^{-j\phi} \sum_{k=-\infty}^{\infty} h[k]\, e^{-j\omega_0 (n - k)} \]

4. Simplifying Each Sum

We can factor out \(e^{j\omega_0 n}\):

\[ y_1[n] = e^{j\phi} e^{j\omega_0 n} \sum_{k=-\infty}^{\infty} h[k]\, e^{-j\omega_0 k} \]

Let \[ H_1 = \sum_{k=-\infty}^{\infty} h[k]\, e^{-j\omega_0 k} \]

Then, \[ y_1[n] = e^{j\phi} H_1 e^{j\omega_0 n} \]

Similarly, \[ y_2[n] = e^{-j\phi} H_1^* e^{-j\omega_0 n} \]

5. Combine the Two Parts

The total output becomes:

\[ y[n] = \frac{A}{2}\left[e^{j\phi} H_1 e^{j\omega_0 n} + e^{-j\phi} H_1^* e^{-j\omega_0 n}\right] \]

If we write \(H_1\) in polar form: \[ H_1 = |H_1| e^{j\theta} \]

Then, \[ y[n] = A |H_1| \cos(\omega_0 n + \phi + \theta) \]

6. Interpretation in Time Domain

• The output remains sinusoidal with the same frequency \(\omega_0\).
• Its amplitude is scaled by \(|H_1|\).
• Its phase is shifted by \(\theta\).

\[ \boxed{y[n] = A |H_1| \cos(\omega_0 n + \phi + \theta)} \]

Introduction

• Signals can be analyzed in both time domain and frequency domain.
• The frequency content of a signal describes how different frequency components contribute to the overall signal.
• Applications in biomedical signals, audio processing, communications, and image processing.

Convolution in Time Domain

• Convolution is a fundamental operation in signal processing.
• Given two signals\(x(t)\)and\(h(t)\), their convolution is defined as:

\[y(t) = x(t) * h(t) = \int_{-\infty}^{\infty} x(\tau) h(t - \tau) d\tau\]

• In discrete-time, convolution is:

\[y[n] = \sum_{k=-\infty}^{\infty} x[k] h[n-k]\]

Convolution Theorem

• Convolution in time domain corresponds to multiplication in frequency domain:

\[X(f) H(f) = Y(f)\]

• This property is crucial in filter design and system analysis.

Introduction to Fourier Series

(-1.0, 4.0)

(-1.0, 4.0)

Introduction to Fourier Series

• Convolution requiere the representation of the signal in a sum of impulse functions.
• Fourier series represents periodic signals as a sum of sinusoids:

\[x(t) = \sum_{n=-\infty}^{\infty} C_n e^{jn\omega_0 t}\]

where\(C_n\)are the Fourier coefficients.

• Decomposing a signal into sinusoidal components allows frequency analysis.

Fourier Coefficients

• The Fourier coefficients\(C_n\)are computed as:

\[C_n = \frac{1}{T} \int_{0}^{T} x(t) e^{-jn\omega_0 t} dt\]

• Determines how much of each frequency is present in the signal.

Example of Fourier Series Expansion

1. Definition of the Complex Fourier Series

For a periodic signal \(x(t)\) with period \(T\):

\[ x(t) = \sum_{k=-\infty}^{\infty} c_k e^{j k \Omega_0 t}, \]

where:

\[ c_k = \frac{1}{T} \int_{T_0}^{T_0 + T} x(t)e^{-jk\Omega_0 t}\,dt, \quad \Omega_0 = \frac{2\pi}{T}. \]

2. Example Signal

We define a periodic signal \(x(t)\) with period \(2\pi\):

\[ x(t) = \begin{cases} 0, & -\pi \le t < 0,\\ 1, & 0 \le t < \pi. \end{cases} \]

This corresponds to a 50% duty cycle pulse.

3. Symbolic Derivation using SymPy

Piecewise((I*(-1 + exp(-I*pi*k))/(2*pi*k), (k > 0) | (k < 0)), (1/2, True))

Expected result:

\[ c_k = \begin{cases} \dfrac{1}{2}, & k=0,\\[4pt] \dfrac{j\left((-1)^k - 1\right)}{2\pi k}, & k \ne 0. \end{cases} \]

4. Numerical Evaluation (NumPy)

import numpy as np
import pandas as pd

def ck_numeric(k):
    k = np.asarray(k, dtype=float)
    return np.where(
        k != 0,
        1j*(np.exp(1j*np.pi*k)-1)/(2*np.pi*k),
        0.5
    )

K = 10
ks = np.arange(-K, K+1)
cks = ck_numeric(ks)

pd.DataFrame({
    "k": ks,
    "Re{c_k}": np.real(cks),
    "Im{c_k}": np.imag(cks),
    "|c_k|": np.abs(cks)
})

     k       Re{c_k}   Im{c_k}         |c_k|
0  -10  1.949086e-17 -0.000000  1.949086e-17
1   -9 -1.949086e-17  0.035368  3.536777e-02
2   -8  1.949086e-17 -0.000000  1.949086e-17
3   -7 -1.949086e-17  0.045473  4.547284e-02
4   -6  1.949086e-17 -0.000000  1.949086e-17
5   -5 -1.949086e-17  0.063662  6.366198e-02
6   -4  1.949086e-17 -0.000000  1.949086e-17
7   -3 -1.949086e-17  0.106103  1.061033e-01
8   -2  1.949086e-17 -0.000000  1.949086e-17
9   -1 -1.949086e-17  0.318310  3.183099e-01
10   0  5.000000e-01  0.000000  5.000000e-01
11   1 -1.949086e-17 -0.318310  3.183099e-01
12   2  1.949086e-17  0.000000  1.949086e-17
13   3 -1.949086e-17 -0.106103  1.061033e-01
14   4  1.949086e-17  0.000000  1.949086e-17
15   5 -1.949086e-17 -0.063662  6.366198e-02
16   6  1.949086e-17  0.000000  1.949086e-17
17   7 -1.949086e-17 -0.045473  4.547284e-02
18   8  1.949086e-17  0.000000  1.949086e-17
19   9 -1.949086e-17 -0.035368  3.536777e-02
20  10  1.949086e-17  0.000000  1.949086e-17

5. Partial Sum Reconstruction

import numpy as np
import matplotlib.pyplot as plt

def x_numeric(t):
    # Original pulse over [-π, π): 0 on [-π,0), 1 on [0,π).
    tw = ((t + np.pi) % (2*np.pi)) - np.pi
    return np.where((tw >= 0) & (tw < np.pi), 1.0, 0.0)

def partial_sum(t, N):
    ks = np.arange(-N, N+1)
    cks = ck_numeric(ks)
    return np.sum(cks[:, None] * np.exp(1j*np.outer(ks, t)), axis=0)

tgrid = np.linspace(-2*np.pi, 2*np.pi, 4000, endpoint=False)
xg = x_numeric(tgrid)

6. Reconstruction for N = 5

N1 = 5
sN1 = partial_sum(tgrid, N1).real

plt.figure(figsize=(8,3))
plt.plot(tgrid, xg, label='x(t) original')
plt.plot(tgrid, sN1, label=f'N={N1}')
plt.xlim([-2*np.pi, 2*np.pi])

(-6.283185307179586, 6.283185307179586)

plt.legend()
plt.title("Partial Sum Reconstruction (N=5)")
plt.show()

7. Reconstruction for N = 20

(-6.283185307179586, 6.283185307179586)

8. Discussion

• The average value \(c_0 = 0.5\) matches the mean level of the pulse.
• As \(N\) increases, the reconstruction converges except near discontinuities.
• The Gibbs phenomenon appears at the jump discontinuities.
• The error \(\lVert x(t)-S_N(t)\rVert_2\) decreases with \(N\).

9. Summary

Example 2 of Fourier Series

(array([-5.,  0.,  5., 10., 15., 20., 25., 30., 35.]), [Text(-5.0, 0, '−5'), Text(0.0, 0, '0'), Text(5.0, 0, '5'), Text(10.0, 0, '10'), Text(15.0, 0, '15'), Text(20.0, 0, '20'), Text(25.0, 0, '25'), Text(30.0, 0, '30'), Text(35.0, 0, '35')])

(array([4.5, 5. , 5.5, 6. , 6.5, 7. , 7.5, 8. , 8.5]), [Text(0, 4.5, '4.5'), Text(0, 5.0, '5.0'), Text(0, 5.5, '5.5'), Text(0, 6.0, '6.0'), Text(0, 6.5, '6.5'), Text(0, 7.0, '7.0'), Text(0, 7.5, '7.5'), Text(0, 8.0, '8.0'), Text(0, 8.5, '8.5')])

Example 2 of Fourier Series

(array([-40., -30., -20., -10.,   0.,  10.,  20.,  30.,  40.]), [Text(-40.0, 0, '−40'), Text(-30.0, 0, '−30'), Text(-20.0, 0, '−20'), Text(-10.0, 0, '−10'), Text(0.0, 0, '0'), Text(10.0, 0, '10'), Text(20.0, 0, '20'), Text(30.0, 0, '30'), Text(40.0, 0, '40')])

(array([4.5, 5. , 5.5, 6. , 6.5, 7. , 7.5, 8. , 8.5]), [Text(0, 4.5, '4.5'), Text(0, 5.0, '5.0'), Text(0, 5.5, '5.5'), Text(0, 6.0, '6.0'), Text(0, 6.5, '6.5'), Text(0, 7.0, '7.0'), Text(0, 7.5, '7.5'), Text(0, 8.0, '8.0'), Text(0, 8.5, '8.5')])

Linearity

• If\(f_1(x)\)and\(f_2(x)\)have Fourier series,
• Then for any constants\(a, b\), -\(a f_1(x) + b f_2(x)\)has a Fourier series,
• With coefficients scaled accordingly.

Time Shifting

• If\(f(x)\)has Fourier coefficients\(a_n, b_n\),
• Then\(f(x - x_0)\)has coefficients: -\(a_n \cos(n\omega x_0) + b_n \sin(n\omega x_0)\),
• And\(b_n \cos(n\omega x_0) - a_n \sin(n\omega x_0)\).

Frequency Scaling

• If\(g(x) = f(cx)\),
• Then the period scales by\(c\),
• The fundamental frequency changes to\(c\omega\),
• Fourier coefficients adjust accordingly.

** Differentiation Property**

• If\(f(x)\)is differentiable,
• Then\(f'(x)\)has Fourier series,
• With coefficients scaled as\(n a_n, n b_n\),
• Higher frequencies get amplified.

Integration Property

• If\(f(x)\)has a Fourier series,
• Then\(\int f(x) dx\)has a Fourier series,
• With coefficients scaled as\(\frac{a_n}{n}, \frac{b_n}{n}\),
• Lower frequencies get emphasized.

Parseval’s Theorem

• The total signal energy is conserved,
• Energy in time domain equals energy in frequency domain,
• Given by: -\(\sum (a_n^2 + b_n^2) = \frac{1}{T} \int |f(x)|^2 dx\).

Convolution Property

• Convolution in time domain,
• Is multiplication in Fourier series coefficients,
• If\(f_1\)and\(f_2\)are convoluted,
• Their Fourier coefficients multiply component-wise.

Discrete Time Fourier Series

• Represents periodic discrete signals using harmonics.
• Extends Fourier series to discrete-time domain.
• Fundamental in digital signal processing.
• Basis for the Discrete Fourier Transform (DFT).

Mathematical Expression

• A periodic sequence\(x[n]\)can be expressed as: -\[x[n] = \sum_{k=0}^{N-1} C_k e^{j(2\pi k n / N)}\].
• The coefficients\(C_k\)are computed as: -\(C_k = \frac{1}{N} \sum_{n=0}^{N-1} x[n] e^{-j(2\pi k n / N)}\).

Periodicity and Symmetry

• The coefficients\(C_k\)repeat every\(N\).
• Ensures correct reconstruction of signals.
• Explains frequency domain representation.
• Basis for spectral analysis.

Key Properties

• Linearity: Superposition holds.
• Time Shift: Causes phase shift in coefficients.
• Parseval’s Theorem: Energy conservation.
• Convolution: Time convolution → Frequency multiplication.

Frequency Domain Interpretation

-\(C_k\)represents discrete frequency content. - The spectrum consists of\(N\)harmonics. - Resolution improves with larger\(N\). - Essential for analyzing periodic discrete signals.

Comparison with Continuous Case

• DTFS applies to discrete periodic signals.
• Continuous Fourier series applies to continuous functions.
• Both represent signals as sums of sinusoids.
• DTFS is used in digital communications and audio processing.

Example of th DTFS

DTFS Coefficients:

C[0] = 2.0000+0.0000j
C[1] = -0.6036-0.6036j
C[2] = 0.0000+0.0000j
C[3] = 0.1036-0.1036j
C[4] = 0.0000+0.0000j
C[5] = 0.1036+0.1036j
C[6] = 0.0000+0.0000j
C[7] = -0.6036+0.6036j

Example 02

Conceptual Foundation

• Fourier Series represents periodic signals in terms of sinusoids.
• As period\(T \to \infty\), the signal becomes aperiodic.
• The Fourier Transform generalizes Fourier Series to aperiodic signals.
• Transforms signals from time to frequency domain.

Mathematical Transition

• Fourier Series of a periodic signal: -\[f(x) = \sum_{n=-\infty}^{\infty} C_n e^{j(2\pi n x / T)}\].
• As\(T \to \infty\), frequency spacing\(\frac{1}{T}\)→ differential.
• Leads to the Fourier Transform: -\[F(\omega) = \int_{-\infty}^{\infty} f(x) e^{-j\omega x} dx\].

Frequency Spectrum Interpretation

• Fourier Series: discrete frequency spectrum.
• Fourier Transform: continuous frequency spectrum.
• Coefficients\(C_n\)become the function\(F(\omega)\).
• Allows analysis of arbitrary signals in frequency domain.

Inverse Fourier Transform

• Recovers time-domain signal from\(F(\omega)\).
• Defined as: -\[f(x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{j\omega x} d\omega\].
• Ensures complete information preservation.
• Basis for signal reconstruction in DSP.

Energy and Parseval’s Theorem

• Energy conservation in time and frequency domains.
• Parseval’s theorem states: -\[\int |f(x)|^2 dx = \frac{1}{2\pi} \int |F(\omega)|^2 d\omega\]
• Ensures no energy loss between domains.